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By adapting the techniques of Finite Element Analysis, the investigator has developed a very general scheme for solving multi-dimensional, non-linear, time-dependent astrophysical boundary value problems in complex geometries. Applications of these techniques include, but are not limited to, the structure and evolution of magnetized rotating stars and planets, interacting binary stars, magnetized thin and thick accretion disks, and full four dimensional solutions to Einstein's equations. Our fully-implicit simulation code solves boundary value equations in up to four (4) dimensions with second or third order accuracy in space and time. The only constraint on the grid in our code is that it be topologically (hyper-)rectangular. Therefore, the grid geometry can be rather arbitrary, making it possible for grid points to lie on complex-shaped core-halo interfaces, photospheres, etc. All coordinate transformations between the grid and physical spaces are determined and performed internally in the code. These properties of the method allow radiative boundary conditions to be applied on grid boundaries (even ones with evolving shape), just as they have been in one-dimensional stellar models in the past. In addition, the local grid spacing and geometry can be adapted to the (multi-dimensional) solution in order to obtain higher accuracy in regions of steep gradients. Any motion of this adaptive grid with respect to the local medium is automatically taken into account by the method in advective derivatives.
The basic Finite Element Method will be discussed along with the modifications necessary for solving astrophysical problems and the steps necessary for adapting this technique to massively-parallel supercomputers, such as the Intel Paragon at Caltech and the Cray T3D at JPL. Some solutions of simple multi-dimensional boundary value problems with and without adaptive gridding will be presented. Recent results on more complex astrophysical problems also will be discussed, pending availability.
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